Claims

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1934_10_01_erdos: Erdős gives a short proof of the theorem of Sylvester and Schur that among any k consecutive integers above k one has a prime factor greater than k, which is the upper bound f(k) at most k.

1955_01_01_erdos: Erdős proves that every block of c k over log k consecutive integers above k, for some constant c greater than 1, contains a multiple of a prime greater than k.

1973_01_01_jutila_ramachandra_shorey: Jutila, Ramachandra and Shorey bound f(k) by a constant times k log log log k over log k log log k, the record upper bound, as Erdős's 1976 survey attests it.

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